The Newton polytope and Lorentzian property of chromatic symmetric functions

TL;DR

The paper investigates chromatic symmetric functions $X_G(oldsymbol{x})$ for three graph families—co-bipartite graphs, indifference graphs of Dyck paths, and incomparability graphs of $(3+1)$-free posets—through their Newton polytopes. It shows these polynomials are SNP and their Newton polytopes are explicit permutahedra $ ext{P}^{(k)}_{oldsymbol{ u}}$, with $oldsymbol{ u}$ given by the greedy coloring weight or dominant colorings. A central contribution is proving the Lorentzian property for abelian Dyck paths and formulating conjectures for all Dyck paths, linking discrete convexity to continuous log-concavity. The work also establishes reductions to unit interval orders, analyzes complexity of coefficient computation (with $ ext{ exttt{ ext#P}}$-completeness in key cases), and connects to representation-theoretic structures via the $ ext{zeta}$ map, highlighting broad interactions between combinatorics, geometry, and algebra. Overall, the results provide a unified SNP/permutahedron framework across three major families, with substantial implications for positivity, convexity, and algorithmic aspects in chromatic symmetric functions.

Abstract

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley--Stembridge conjecture, we show that the allowable coloring weights for indifference graphs of Dyck paths are the lattice points of a permutahedron $\mathcal{P}_λ$, and we give a formula for the dominant weight $λ$. Furthermore, we conjecture that such chromatic symmetric functions are Lorentzian, a property introduced by Brändén and Huh as a bridge between discrete convex analysis and concavity properties in combinatorics, and we prove this conjecture for abelian Dyck paths. We extend our results on the Newton polytope to incomparability graphs of (3+1)-free posets, and we give a number of conjectures and results stemming from our work, including results on the complexity of computing the coefficients and relations with the $ζ$ map from diagonal harmonics.

Figures (11)

• Figure 1: A Dyck path $d$ encoded by the Hessenberg function $h_d=(3,3,5,5,5)$ and its indifference graph $G(d)$ which is an incomparability graph of a unit interval order poset $P$.
• Figure 2: The Newton polytopes of $X_{G(d)}(x_1,x_2,x_3)$ and $X_{G(d)}(x_1,x_2,x_3,x_4)$ for $d=\mathsf{n} \mathsf{n} \mathsf{n} \mathsf{e} \mathsf{e} \mathsf{n} \mathsf{n} \mathsf{e} \mathsf{e} \mathsf{e}$ are the permutahedra $\mathcal{P}_{221}^{(3)}$ and $\mathcal{P}^{(4)}_{221}$, respectively.
• Figure 3: Description of bounce path algorithm to determine the greedy coloring weight $(2,2,1)$.
• Figure 4: (A) A part listing $L$ and (B) its corresponding (3+1)-free poset $P$.
• Figure 5: The part listings $L_0$, $L_1$, and $L_2$ in the convex combination of $X(L)$. The dominant coloring $\kappa_2$ of $X(L_2)$ dominates the respective dominant colorings $\kappa_0$ and $\kappa_1$ of $X(L_0)$ and $X(L_1)$.

Theorems & Definitions (91)

• Conjecture 1.1: Stanley StChrom1
• Conjecture 1.2: Stanley--Stembridge StSt
• Theorem 1.3
• Conjecture 1.4: Conjecture \ref{['conj:dyckLor']}
• Theorem 1.5: Theorem \ref{['thm: Lorentzian abelian case']}
• Conjecture 1.6: Monical Mphd
• Theorem 1.7
• Theorem 2.1: Rado Rado
• Remark 2.2
• Example 2.3